What is it called in AI when a program is designed to make "x in the style of y"?
This problem is known as "style transfer". 
The field was started by Gatsys et al. in 2016, and has seen a lot of work over the last few years, including conditional translation from Isola et al. in 2017 and unpaired translation from Zhu et al. 2017.

This picture (Figure 2 from Gatsys et al.) shows the idea behind style transfer, and illustrates the dramatic results that are possible with these models.